Entropy growth during free expansion of an ideal gas

To illustrate Boltzmann's construction of an entropy function that is defined for a microstate of a macroscopic system, we present here the simple example of the free expansion of a one dimensional gas of non-interacting point particles. The construction requires one to define macrostates, corresponding to macroscopic variables. We define a macrostate $M$ by specifying the fraction of particles in rectangular boxes $\Delta x \Delta v$ of the single particle position-velocity space $\{x,v\}$. We verify that when the number of particles is large the Boltzmann entropy, $S_B(t)$, of a typical microstate of a nonequilibrium ensemble coincides with the Gibbs entropy of the coarse-grained time-evolved one-particle distribution associated with this ensemble. $S_B(t)$ approaches its maximum possible value for the dynamical evolution of the given initial state. The rate of approach depends on the size of $\Delta v$ in the definition of the macrostate, going to zero at any fixed time $t$ when $\Delta v \to 0$. Surprisingly the different curves $S_B(t)$ collapse when time is scaled with $\Delta v$ as: $t \sim \tau/\Delta v$. We find an explicit expression for $S_B(\tau)$ in the limit $\Delta v \to 0$. We also consider a different, more hydrodynamical, definition of macrostates for which $S_B(t)$ is monotone increasing, unlike the previous one which has small decaying oscillations near its maximum value. Our system is non-ergodic, non-chaotic and non-interacting; our results thus illustrate that these concepts are not as relevant as sometimes claimed, for observing macroscopic irreversibility and entropy increase. Rather, the notions of initial conditions, typicality, large numbers and coarse-graining are the important factors. We demonstrate these ideas through extensive simulations as well as analytic results.